Laying the Foundation

Einstein's Theories of Relativity and Gravitation by Albert Einstein, is part of the HackerNoon Books Series. You can jump to any chapter in this book here. Laying the Foundation

Laying the Foundation

The modern geometer falls in with Euclid when he writes an elementary text, satisfying the beginner’s demand for apparent rigor by defining point [116]and line in some fashion. But when he addresses to his peers an effort to clarify the foundations of geometry to a further degree of rigor and lucidity than has ever before been attained, he meets these difficulties from another quarter. In the first place he is always in search of the utmost possible generality, for he has found this to be his most effective tool, enabling him as it does to make a single general statement take the place and do the work of many particular statements. The classical geometer attained generality of a sort, for all his statements were of any point or line or plane. But the modern geometer, confronted with a relation that holds among points or between points and lines, at once goes to speculating whether there are not other elements among or between which it holds. The classical geometer isn’t interested in this question at all, because he is seeking the absolute truth about the points and lines and planes which he sees as the elements of space; to him it is actually an object so to circumscribe his statements that they may by no possibility refer to anything other than these elements. Whereas the modern geometer feels that his primary concern is with the fabric of logical propositions that he is building up, and not at all with the elements about which those propositions revolve.

It is of obvious value if the mathematician can lay down a proposition true of points, lines and planes. But he would much rather lay down a proposition true at once of these and of numerous other things; for such a proposition will group more phenomena under a single principle. He feels that on pure [117]scientific grounds there is quite as much interest in any one set of elements to which his proposition applies as there is in any other; that if any person is to confine his attention to the set that stands for the physicist’s space, that person ought to be the physicist, not the geometer. If he has produced a tool which the physicist can use, the physicist is welcome to use it; but the geometer cannot understand why, on that ground, he should be asked to confine his attention to the materials on which the physicist employs that tool.

It will be alleged that points and lines and planes lie in the mathematician’s domain, and that the other things to which his propositions may apply may not so lie—and especially that if he will not name them in advance he cannot expect that they will so lie. But the mathematician will not admit this. If mathematics is defined on narrow grounds as the science of number, even the point and line and plane may be excluded from its field. If any wider definition be sought—and of course one must be—there is just one definition that the mathematician will accept: Dr. Keyser’s statement that “mathematics is the art or science of rigorous thinking.”

The immediate concern of this science is the means of rigorous thinking—undefined terms and definitions, axioms and propositions. Its collateral concern is the things to which these may apply, the things which may be thought about rigorously—everything. But now the mathematician’s domain is so vastly extended that it becomes more than ever important for him to attain the utmost generality in all his pronouncements.[118]

One barrier to such generalization is the very name “geometry,” with the restricted significance which its derivation and long usage carry. The geometer therefore must have it distinctly understood that for him “geometry” means simply the process of deducing a set of propositions from a set of undefined primitive terms and axioms; and that when he speaks of “a geometry” he means some particular set of propositions so deduced, together with the axioms, etc., on which they are based. If you take a new set of axioms you get a new geometry.

The geometer will, if you insist, go on calling his undefined terms by the familiar names “point,” “line,” “plane.” But you must distinctly understand that this is a concession to usage, and that you are not for a moment to restrict the application of his statements in any way. He would much prefer, however, to be allowed new names for his elements, to say “We start with three elements of different sorts, which we assume to exist, and to which we attach the names A, B and C—or if you prefer, primary, secondary and tertiary elements—or yet again, names possessing no intrinsic significance at all, such as ching, chang and chung.” He will then lay down whatever statements he requires to serve the purposes of the ancient axioms, all of these referring to some one or more of his elements. Then he is ready for the serious business of proving that, all his hypotheses being granted, his elements A, B and C, or I, II and III, or ching, chang and chung, are subject to this and that and the other propositions.[119]

The objection will be urged that the mathematician who does all this usurps the place of the logician. A little reflection will show this not to be the case. The logician in fact occupies the same position with reference to the geometer that the geometer occupies with reference to the physicist, the chemist, the arithmetician, the engineer, or anybody else whose primary interest lies with some particular set of elements to which the geometer’s system applies. The mathematician is the tool-maker of all science, but he does not make his own tools—these the logician supplies. The logician in turn never descends to the actual practice of rigorous thinking, save as he must necessarily do this in laying down the general procedures which govern rigorous thinking. He is interested in processes, not in their application. He tells us that if a proposition is true its converse may be true or false or ambiguous, but its contrapositive is always true, while its negative is always false. But he never, from a particular proposition “If A is B then C is D,” draws the particular contrapositive inference “If C is not D then A is not B.” That is the mathematician’s business.

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